Image restoration by deconvolution

Volker Bäcker

Montpellier Rio Imaging http://www.mri.cnrs.fr/

Pierre Travo

IFR3Giacomo Cavalli

Frederic BantigniesPatrice Mollard

Nicole Lautrédou-Audouy Jean-Michel Poulin

volker.baecker@crbm.cnrs.fr

Overview

➲ Part 1● introduction

● what is deconvolution ?

● how does it work ?

● when should it be used ?

➲ Part 2● what are the parameters to know and care about

for image restoration by deconvolution?

fluorescence microscopy

● fluorescence microscopy●specimen marked with dye that emists light of one wavelength while being stimulated by light of another wavelength

●Microscope types●widefield whole specimen bathed in light●confocal image is constructed point by point to keep out out-of-focus light●two photon two photons needed to stimulate emission, similar effect as confocal

● specimen has to be in focal distance

● to image 3d specimen

●move focal plane through specimen

●creating stack of slides

Example: 2d widefield

Immunostaining on whole mount drosophila EmbryoUsing an antibody against a nuclear protein

Image from microscope After deconvolution(same levels)

Example 3d confocal

After deconvolutionImage from microscope

Example: time series 2 photon

Image from microscope After deconvolution

The aquired image is not the „real“ image

➲ Images are degraded due to the limited aperture of the objective

➲ Deconvolution can be used to get an image nearer to the real object

● by using knowledge of the imaging process and the properties of the microscope

➲ Deconvolution can be used for all kinds of fluorescence microscope images:

● 2D, 3D, time series, widefield, confocal, 2 photon

Sources of image degradation

➲ Noise

➲ Blur

● Can be handled by image restoration

➲ Scatter

● random distribution of light due to heterogenous refrection index within specimen

➲ Glare

● random distribution of light that occurs within the optical train of the microscope

Causes of image degradationNoise

➲ Geben Sie eine Zusammenfassung der momentanen Situation

Causes of image degradationNoise

➲ Where does the noise come from ?

● random fluctuations in the signal intensity

● variation of the incident photon flux

● interfering signals from electronic system of the captor device

Causes of image degradation Blur

Before restoration

After restoration

Causes of image degradationBlur

➲ Where does the blur come from ?

● contributions of out-of-focus light to the imaging plane

● diffraction

● a result of the interaction of light with matter

● diffraction is the bending of light as it passes the edge of an object

How does deconvolution work

➲ Image restoration

● Get rid of noise

● assume random noise with Poisson distribution

● remove it

● Get rid of blur

● Compute real image from sample

● by applying a model of how the microscope degraded the image

deconvolution

Point Spread Function➲ Point spread function (psf)

● Model of how one pointis imaged by microscope

● Experimental

● aquired by taking an image of„point like objects“ - beads

● Alternatevely, point like objectpresent in the acquired imageitself can be usedf.

● Theoretical

● computed from the microscopeand captor parameters

Convolution (Faltung)

➲ aquired image = real image convolved with psf

➲ Convolution is an integral that expresses

● amount of overlap of functions as g is shifted over f.

● N pixel => O(N*N) operations to compute it

i x f x ' g x x ' dx '

i(x) : aquired imagef(x) : object functiong(x) : point spread function

Fourier Transform (FT)

F f x e i 2 x dx

● Signal can be represented as sum of sinoids

● FT transforms from spacial to frequency domain

Convolution theorem

i(x) : aquired imagef(x) : object functiong(x) : point spread

function

I fourier transform of iF fourier transform of fG fourier transform of g

i x f x ' g x x ' dx ' I F G<=>

*

inverse FTFourier transform (FT) FT

Object function convolved with psf

Object function psf

FT can be computed inO(n * log n)

Deconvolution

➲ Deconvolution: find object function f for given image i and psf g

➲ Unfortunatly it is not practicable to compute

● G has zeros outside certain regions● Setting F zero for these would create artefacts

● In practice there is noise● N/G would amplify noise

➲ It's not possible to reconstruct the real object function

i x f x ' g x x ' dx ' I F G<=>

FIG

I F G N

Deconvolution algorithms

➲ Solution

● Find an algorithm that computes a function f' so that

● f' estimates f as good as possible

● works in the presence of noise

➲ Different deconvolution algorithms exist

➲ In general best for fluorescent microscopy:

● (Classical) Maximum Liklihood Estimation - MLE

Maximum Likelihood Estimation

➲ Tries to optimise f' iteratively

➲ The basic principal is (but there's more to it)

➲ g(i|j) : psf - the fraction of light from true location j that is observed in pixel i

f new , j f old , ji

iig i j

k

g j k f old , k

Fraction of light from pixel j that hit other pixels

Fraction of light from other pixelsthat hit pixel j

Richardson and LucyR-L Iteration

6 5

3 4 4 3

0,3

1

0,1

0,20,2

5 * [5*1 / (5*1 + 0.3*4 + 0.2*6) + 0,1*4 /(5*1 + 0.3*4 + 0.2*6) + 0,2*6/(5*1 + 0.3*4 + 0.2*6)]

5 * [5 / 7.4 + 0.4/7.4 + 1.2/7.4]

5 * [(5 + 0.4 + 1.2)/7.4]

5 * [6.6 /7.4]

5 * 0.891891

4.459459

f new , j f old , ji

iig i j

k

g j k f old , k

psf

image

1

2

3

4

A B C D

Denominator:get rid of foreign light that hit me 1 C3 + 0.3 C4 + 0.2 B3

Numeratorrealign my light to me

1 C3 + 0.1 C4 + 0.2 B3

New estimate

last estimate

aquired image

fraction of light lost

fraction of light from others

last estimate

✔ image from microscope is degraded

✔ it contains noise and blur

✔ blur can be described as a convolution of object function and psf

✔ image nearer to the object function can be obtained by image restoration yielding higher resolution and better contrast

✔ MLE is a deconvolution algorithm approriate for fluorescent microscope images

➔ imaging process is not finished finished without deconvolution

➔ do it whenever high quality images are needed

Summary and conclusions 1

Image restoration in practice

Many deconvolution software packages are commercially available

They use various types of deconvolution algorithms

In addition to these algorithms, they might incorporate other imaging tools, such as filters of different kinds. Moreover, different types of algorithms may introduce or not some « assumptions » concerning the image sent to restoration.

In general, it is important to test the software. One basic « rule of thumb » is also that the restoration should respect the acquired image in terms of objects visible and of their relative intensity. Objects « appearing », « disappearing » or changing relative intensity with respect to neighboring structures are diagnostic of problems. These problems might be due to the setting of relevant parameters or, in the worst case, of poor quality of the software

Image restoration using the huygens2 software from SVI

➲ http://www.svi.nl/ - website of Scientific Volume Imaging (SVI)➲ It is the software used at the

Institute of Human Genetics

Relevant parameters in deconvolution Setting Microscope parameters

➲ microscope type

● widefield and multipoint confocal ● work with ccd camera

● single point confocal and two photon● work with photomultiplier

● different point spread functions

➲ if you don't know

● Ask your imaging facility and look at the specifications of your microscope

Microscope parameters➲ Numerical aperture

● measure of ability to gather light and resolve fine specimen detail at a fixed object distance

● higher magnification doesn't yield higher resolution, higher NA does

➲ Maximal value written on objective

➲ Can't be larger than the the refractive index n of the medium

NA nsin

Sampling theorem

● Imaging converts an anlog signal into a digital signal

● When converting an analog signal into a digital signal the sampling theorem applies

Nyquist-Shannon sampling theorem“the sampling interval must not be greater than one-half the size of

the smallest resolvable feature of the optical image”

● sampling at nyquist rate means using exactly this interval

● sampling interval is the pixel size in our image

x 4 NA

Undersampling and oversampling

under sampling●loss of information●aliasing artefacts

over sampling●higher computation times and storage requirements●longer acquisition times, photobleaching.

under sampling example. An object of a given shape (dashed line) can be interpreted as a different shape (thick line) if too few points are acquired along any of the x,y,z axes

Changing the Numerical Aperture (NA) forwidefield / two photon

➲ huygens2 allows under/oversampling within a range

➲ at the borders of this range deconvolution can be done but results are not good

➲ In this case better results when “lying” about NA

➲ if sampling size not in range change NA

0.5i

x x 1.2i

x x 4NA

i

x

nyquist sample size

Microscope parameters

➲ Excitation and emission wavelength

➲ fluorescent dye absorbs light of one wavelength and emits light of another wavelength

➲ filter cubes are used to ensure that only light of a wanted wavelength passes.

● exitation and emission wavelengths depend on the cube used

● GFP 473, 525

Microscope parameters

➲ The objective magnification used

● determines the pixel size in the image

● ccd camera

● Pixel size = ccd element size / magnification(eventually modified by other parameters)

● photomultiplier

● pixel size depends on resolutionand magnification

Microscope parameters

➲ Refractive index n of the objective medium

● oil 1,51500

● water 1,33810

● air 1,00000

➲ Should match the refractive index of the sample medium

● Otherwise

● Magnification error in axial direction

● Spherical aberration (psf deteriorates with increasing depth)

Microscope parameters

➲ Cmount factor

● adaptor that attaches the camera to the microscope

● might contain additional optic that

● changes the overall magnification

● and therefore the pixel size

● value is 1 if no additional optic present

psccd

obm cmf

Microscope parameters

➲ Tube factor

● the tube might contain additional optics to change the tube length

● this changes the overallmagnification

● and therefore the pixel size

psccd

obm cmf tf

Microscope parameters

➲ sample medium

➲ refractive index n

● default (all media for example water) 1,33810

● liquid Vectashield (not polymerized) 1,49000

● 90-10 (v:v) glycerol - PBS ph 7.4 1,49000

● prolong antifade 1,4

➲ limits the NA and therefore the possible resolution

NA nsin

Captor parameters

➲ size of the unitary ccd captor

➲ image sensor of the camera

● ccd – charge coupled device

● diodes that convert light into electrical charge

➲ property of the camera

● Coolsnap 6450 nm

● Micromax 6700 nm

For photomultiplier● the pixel size is asked● see table in help pages

Captor parameters

➲ Binning

➲ take nxn elements as one

➲ more light per pixel

➲ reduces noise

➲ higher signal to noise ratio

➲ lower resolution

psccd bin

obm cmf tf

Captor parameters

➲ in case of XZY

● z step size

➲ in case of time series

● time interval

Captor parameters

➲ in case of confocal

● pinhole radius

● pinhole

● keep out out of focus light

● pinhole either fixed or adjustable

● Backprojected radius in nm

● Size of pinhole as it appears in the specimen plane

● size should match airy disk (2d psf) size

rbr phys

msystemmobj

6.66 for LSM510

task parameter➲ Style of processing

● step

● process image slide by slide

● converts stack into time series for processing

● converts result back into stack

● volume

● use 3d information

● step combined

● do step processing

● followed by volume processing with fixed parameters

Full restoration parameters

➲ signal/noise ratio

● the ratio of signal intensity to noise intensity

● high noise case

● can be measured in the image

● Single photon hit intensity

● find low intensity voxels from one photon hit – add values – subtract background

● Max voxel value

● value of brightest voxel

● low noise case

● single photon hits can´t be seen

● rough guess is sufficient

SN

maxVoxelValuesinglePhotonHitIntensity

Full restoration parameters

➲ background offset

● empty regions should be black

● but contain some light in reality

● subtract mean background to see object clearly

Full restoration parameters

➲ number of iterations

● too low

● optimal restoration not yet achieved

● too high

● takes longer to compute

● some signal may be removed

● Usually between 30-70

f new , j f old , ji

iig i j

k

g j k f old , k

Summary and conclusions 2

➲ deconvolution should be used to obtain high quality imagesfor all kind of fluorescent microscope images

➲ parameters of the imaging system have to be entered to create a model of the image degradation

End of presentation

volker.baecker@crbm.cnrs.frGiacomo.Cavalli@igh.cnrs.fr

Links

participants➲ Montpellier RIO Imaging

http://www.mri.cnrs.fr/➲ IFR3 / CCIPE

http://www.montp.inserm.fr/ifr3.htm➲ IGH

http://www.igh.cnrs.fr/➲ CRIC

http://www.iurc.montp.inserm.fr/cric/index.htm

literature● Introduction to Fluorescence Microscopy

http://www.microscopyu.com/articles/fluorescence/fluorescenceintro.html● How does a confocal microscope work?

http://www.physics.emory.edu/~weeks/confocal/● Two-Photon Fluorescence Microscopy

http://www.fz-juelich.de/ibi/ibi-1/Two-Photon_Microscopy/● Deconvolution in Optical Microscopy

http://micro.magnet.fsu.edu/primer/digitalimaging/deconvolution/deconintro.html

Links

literature● Diffraction of Light

http://micro.magnet.fsu.edu/primer/java/diffraction/basicdiffraction/● Image restoration: getting it right

http://www.svi.nl/support/talks/GettingItRight.pdf● Image Restoration in Fluorescence Microscopy

http://www.ph.tn.tudelft.nl/Publications/PHDTheses/GMPvanKempen/thesis_kempen.html● Image restoration in one- and two-photon microscopy

http://www.svi.nl/support/talks/Vancouver97.pdf● Introduction to Convolution

http://cnx.rice.edu/content/m11542/latest/● An Introduction to Fourier Theory

http://aurora.phys.utk.edu/~forrest/papers/fourier/● A Self Contained Introduction to Fourier Transforms

http://www.doc.eng.cmu.ac.th/course/cpe496/notes/fourier.pdf● Convolution theorem

http://www.fact-index.com/c/co/convolution_theorem.html

Links

literature● Three-Dimensional Imaging by Deconvolution Microscopy

Article ID meth.1999.0873, available online at http://www.idealibrary.com on IDEAL● Deconvolution of confocal images of dihydropyridine and ryanodine receptors

in developing cardiomyocyteshttp://www.sfu.ca/~tibbits/research/JAP04.pdf

● Maximum likelihood estimation via the ECM algorithm: A general frameworkhttp://www.jbs.agrsci.dk/~lfo/talks/ECM_talk.pdf

● The influence of the background estimation on the superresolution properties ofnon-linear image restoration algorithmshttp://www.ph.tn.tudelft.nl/People/lucas/publications/1999/SPIE99GKLV/SPIE99GKLV.pdf

● Numerical Aperture and Resolutionhttp://micro.magnet.fsu.edu/primer/anatomy/numaperture.html

● User guide for Huygens Professional and Deconvolution Recipeshttp://www.svi.nl/download/

● Digital Image Sampling Frequencyhttp://www.olympusmicro.com/primer/java/digitalimaging/processing/samplefrequency/index.html

● Filter Cubeshttp://www.olympusmicro.com/primer/techniques/fluorescence/filters.html

● Filters for fluorescence microscopyhttp://www.nikon-instruments.jp/eng/products/option/index1.aspx

Links

literature● Immersion Media

http://www.olympusmicro.com/primer/anatomy/immersion.html● How Digital Cameras Work

http://electronics.howstuffworks.com/digital-camera2.htm● Pixel Binning

http://micro.magnet.fsu.edu/primer/digitalimaging/concepts/binning.html● CCD Signal-To-Noise Ratio

http://www.microscopyu.com/tutorials/java/digitalimaging/signaltonoise/